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6/2/2010 Accounting for Earth Curvature in
Directional Drilling - SPE 96813
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Accounting for Earth Curvature in Directional Drilling
Accounting for Earth Curvature Accounting for Earth Curvature
in Directional Drillingin Directional Drilling
Noel ZinnNoel Zinn
ExxonMobil Exploration CompanyExxonMobil Exploration Company
Society of Petroleum EngineersSociety of Petroleum Engineers
Annual Technical Conference and ExhibitionAnnual Technical Conference and Exhibition
1010--13 October 200513 October 2005
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Homage to a Previous SPE PaperHomage to a Previous SPE Paper
• Directional Drilling and Earth Curvature
• Hugh Williamson and Harry Wilson
• SPE 56013 (1999)
• Williamson and Wilson explain 7
geodetic mistakes in the computation of
deviated-well survey data
• Errors range from centimeters to 10 or
more meters to hundreds of meters
I begin with homage to a previous paper. In an important contribution to the
geodesy of directional drilling, authors Hugh Williamson and Harry Wilson
discuss the causes - and offer cures - for 7 geodetic mistakes found in current
oil field practice. The positional errors resulting from these mistakes range
from a few centimeters to about 10 meters to hundreds of meters.
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Mistakes of Earth CurvatureMistakes of Earth Curvature
1- Different wellhead and target datums
2- Wrong wellhead and target datums
3- Imprecise grid convergence
4- Grid convergence change not accounted for
5- Grid scale factor not accounted for
6- Grid scale factor change not accounted for
7- Depth adjustment not applied
Space Saver
Of the 7 mistakes cited by Williamson and Wilson, the 2 mistakes shown in
white at the top are lapses in geodetic planning that must be prevented by
prudent practice no matter how the survey is executed or computed.
The 5 five mistakes in yellow are of a different character. These are not
planning lapses or unavoidable measurement errors. They are computational
mistakes that can be corrected by changes in software.
Numbers 3, 4, 5 and 6 relate to the use of map projections to convert
displacements (also known as departures) into geodetic coordinates.
Mistakes number 4 and number 7 are common in the software I’ve tested.
These mistakes are not accounting for changes in convergence over the reach of
a well and not adjusting for depth.
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Mistakes of Earth CurvatureMistakes of Earth Curvature
1- Different wellhead and target datums (<1000m)
2- Wrong wellhead and target datums (≈7m)
3- Imprecise grid convergence (<1m)
4- Grid convergence change not accounted for (15m)
5- Grid scale factor not accounted for (10m)
6- Grid scale factor change not accounted for (<1m)
7- Depth adjustment not applied (5.5m for 3500m average TVD)
Well: 60° latitude / 10km extended reach
The 7 mistakes are detailed here with quantified positioning errors shown to
the right for the extreme well detailed at the bottom of the slide. The well is
high-latitude and extended in reach. The seventh mistake is quantified for a
very deep well. These extreme features are within the scope of today’s
technology.
Don't be concerned about the kilometer error shown on the first line. It won't
happen to you and it's not the subject of this paper.
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The SolutionsThe Solutions
• Williamson and Wilson offer depth amplification and map-based applications of grid-north convergence and linear scale factor as the cure for 5 of the 7 mistakes
• This paper offers a simple alternative (LMP) that avoids grid north, requires no mapping corrections, is computationally efficient and equally accurate
Williamson and Wilson offer the application of depth amplification and map-
based corrections for convergence of the meridians and linear scale factor as
the solution for the 5 computational mistakes.
In this paper I take a radically different approach. I offer an alternative
algorithm (called LMP) that converts True Vertical Depth and horizontal
displacements directly into latitude and longitude without an intervening map
projection, without convergence of the meridians and without linear scale
factors. Instead, LMP uses three easily computed radii of the ellipsoidal Earth.
Repeated computation of these radii is more efficient than computing map
corrections. Grid north is avoided, and I’ll have much to say about grid north
in this talk.
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The Four Survey Data TypesThe Four Survey Data Types
• Raw deviated- well survey data
– Native accelerometer, magnetometer, gyro,
inertial and pipe-tally data - lots of it!
• Intermediate data (reduced/exchanged)
– Inclination, azimuth, measured depth, tool face
• Cubical coordinates
– TVD and N/S & E/W departures/displacements
• Geodetic coordinates
– Latitude, Longitude, Northing, Easting
To better understand the problems that LMP solves, I begin with the following
taxonomy of survey data types and the formats in which these data are
delivered.
Raw survey data are those acquired by field instruments – accelerometers,
magnetometers, gyros, and – of course – pipe tally. These observations are
acquired in great quantity and are rarely seen in the operating companies.
Instead, raw data are reduced into intermediate data (inclination, azimuth,
measured depth). Intermediate data is delivered as the permanent record of a
well survey.
Cubical – or engineering - coordinates are True Vertical Depth and
displacements in the N/S and E/W directions.
Geodetic coordinates in the horizontal are either geographical coordinates
(latitude and longitude) or projected coordinates (Northings and Eastings, or
X’s and Y’s). TVD remains unchanged as the vertical coordinate.
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Intermediate Survey Data into Intermediate Survey Data into
Cubical CoordinatesCubical Coordinates
∆∆∆∆ TVD D/U
Disp N/S
Disp E/W
Inc1
∆∆∆∆ MDAz2
Az1
Inc2
If the first computation is the reduction of raw data into its intermediate form,
the second computation is the reduction of intermediate data into cubical
coordinates.
In this graphic our survey stations are at diagonally opposite vertices of the
box. The five measurements are the inclination and azimuth entering the box,
the inclination and azimuth leaving the box and the difference in measured
depth between the two vertices shown as the gentle curve. We must solve for
three cubical coordinates: (1) N/S displacement, (2) E/W displacement and (2)
delta TVD Down/Up. Simple formulas to accomplish this by different methods
(including Minimum Curvature) are given by the API.
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Our project extracted from
a cubical, “flat” earth
Our project extracted
from an ellipsoidal earth
Cubical to Geodetic CoordinatesCubical to Geodetic Coordinates
Because the Earth is not a cube, cubical coordinates are an inadequate
Reference System for important technical and safety issues, such as avoiding
well collisions. Cubical coordinates must be converted into geodetic
coordinates. The usual way to accomplish this is to relate cubical coordinates to
Northings and Eastings determined with a map projection.
This does work, but – as previously mentioned - complete geodetic accuracy
requires the computation and application of several corrections at every station
along the well path. Unfortunately, these corrections are not always computed
and applied. Errors result.
Furthermore, map projections introduce grid north as an azimuth reference in
addition to true north and magnetic north. Grid north is a common cause of
mistakes in the management of survey data.
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Our project extracted from an ellipsoidal earth
Depth AmplificationDepth Amplification
If this graphic appears fuzzy to you, think of it as seismic data!
Surveyors using conventional equipment have historically surveyed from hills,
buildings or towers for better visibility. These surveyors always made a “height
reduction” (or shortening) of the distances observed at these elevations to the
surface of the ellipsoid.
Because of Earth curvature, the vertical lines shown in white on either side of
the graphic are not parallel (as they are in a cube). They tend to meet at the
geocenter. As the elevation changes along two verticals, the horizontal distance
between them changes (as shown in red). The geographical coordinates of the
verticals don't change, just the distance between them.
Similarly, a displacement at depth changes geographical coordinates faster
than the same displacement near the surface.
The seventh mistake cited by Williamson and Wilson is the lack of this
correction.
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Contractor Survey FormatsContractor Survey Formats
• Intermediate data and cubical and geodetic coordinates delivered in a spreadsheet
• Formats specific to survey companies
• Often referenced to grid north– An unfortunate coupling of computational
convenience with field acquisition
– True or magnetic north can be observed with instruments – grid north cannot
– Grid north is a source of data-management confusion
Survey contractors typically deliver intermediate data, cubical coordinates and
geodetic coordinates in a spreadsheet in formats that are specific to the specific
contractors. Cubical coordinates are not the same as projected coordinates.
Some presentations are well documented with respect to this distinction and the
differences among true, magnetic and grid north. Some are not.
As mentioned, the use of a map projection in the conversion of cubical to
geodetic coordinates necessarily introduces grid north as an azimuth reference.
Unlike true or magnetic north, which can be physically observed in the field
with appropriate instruments, grid north is a mathematical abstraction. Grid
north is a consequence of the grid parameters chosen for the projection, either
by convention, for convenience or rather arbitrarily. Change the grid
parameters, especially the central meridian, and grid north changes.
In the operating companies a regional geological study may, indeed, require a
change of grid parameters. Australian map projections reverse the sign
convention for convergence! Grid north can also be found in field acquisition,
for example, free gyros aligned with a platform grid-north reference. But the
interpretation project using the data may be in a different projection
altogether.
These are just some reasons why this confusing coupling of computational
convenience with acquisition and exchange can cause mistakes in the
management of survey data. Grid north is best avoided - if possible.
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Data Format StandardsData Format Standards
• UK Offshore Operators Association
– UKOOA P7/2000: ASCII, intermediate and
geodetic data, geodetically unambiguous.
• Minerals Management Service
– MMS P7/2000: required for US Federal OCS
– Clone of UKOOA P7/2000
• POSC
– WITSML: XML raw acquisition data
– WellPathML: XML version of UKOOA P7 for
intermediate and geodetic data
The UK Offshore Operators Association (UKOOA) has addressed the issue of
inadequate documentation with the UKOOA P7/2000 format. The P7 format is
geodetically unambiguous. It is widely used in Europe.
In the United States the Minerals Management Service (MMS) now requires a
clone of UKOOA P7/2000 (called MMS P7/2000) for wells spud in the Federal
Outer Continental Shelf after July 26th of last year (2004).
POSC (the Petrotechnical Open Standards Consortium) has two formats.
WITSML (Wellsite Information Transfer Standard Markup Language) can
capture raw survey data for exchange.
WellPathML is emerging as an XML version of UKOOA P7 for intermediate,
cubical and geodetic data.
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Declination= +5°(5°E)
G
MT
Azimuth Reference = Grid North
(magnetic) survey azimuth = 11°
Grid azimuth = 11°+ 5°- 2°= 14°
Convergence= +2°(2°E)
Survey
11°
Declination= -7°(7°W)
T
Azimuth Reference = True North
(magnetic) survey azimuth = 28°
True azimuth = 28°- 7°= 21°
Survey
28°
Declination= +5°(5°E)
G
MT
Azimuth Reference = Grid North
(magnetic) survey azimuth = 11°
Grid azimuth = 11°+ 5°- 2°= 14°
Convergence= +2°(2°E)
Survey
11°
Declination= -7°(7°W)
T
GM
Azimuth Reference = True North
(magnetic) survey azimuth = 28°
True azimuth = 28°- 7°= 21°
Survey
28°
From UKOOA Data Exchange Format P7/2000
Excerpt from UKOOA P7 FormatExcerpt from UKOOA P7 Format
One benefit of the UKOOA P7 format and its inheritors is the clarity of
geodetic documentation.
For example, this graphic from the P7 format offers guidance in reconciling
true north, magnetic north, grid north and survey direction.
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Declination = +5 ° ( 5 ° E)
M T
Azimuth Reference = Grid North
(magnetic) survey azimuth = 11 °
Grid azimuth = 11 ° + 5 ° - 2 ° = 14 °
Convergence = +2 ° ( 2 ° E)
Survey
11 °
Declination = - 7 ° ( 7 ° W)
T
Azimuth Reference = True North
(magnetic) survey azimuth = 28 °
True azimuth = 28 ° - 7 ° = 21 °
Survey
28 °
Declination = +5 ° ( 5 ° E)
M T
Azimuth Reference = Grid North
(magnetic) survey azimuth = 11 °
Grid azimuth = 11 ° + 5 ° - 2 ° = 14 °
Convergence = +2 ° ( 2 ° E)
Survey
11 °
Declination = - 7 ° ( 7 ° W)
T
M
Azimuth Reference = True North
(magnetic) survey azimuth = 28 °
True azimuth = 28 ° - 7 ° = 21 °
Survey
28 °
From UKOOA Data Exchange Format P7/2000
UKOOA P7: Edited ExcerptUKOOA P7: Edited Excerpt
Look how simple this graphic would be if we eliminate grid north from the
computations!
It can be done.
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Different Routes fromDifferent Routes from
Cubical to Geodetic CoordinatesCubical to Geodetic Coordinates
1- Intermediate data
2- Displacements / TVD
3- Apply grid convergence and scale corrections
4- Projected X/Y
5- Latitude / Longitude
1- Intermediate data
2- Displacements /
TVD
3- Divide by earth radii
4- Latitude / Longitude
5- Projected X/Y (grid north not required)
Current Route Current Route LMP RouteLMP Route
Considering some of the issues discussed so far, an ExxonMobil colleague
(David M. Lee) proposed that well-path latitude and longitude be computed
directly from cubical coordinates and the radius of the Earth, that is, without a
map projection. I modified David’s proposal for an ellipsoidal Earth with an
infinite number of radii of curvature that vary as functions of latitude and
azimuth. The result is Lee’s Modified Proposal, or LMP.
The current route to geodetic coordinates is that on the left. All the map
corrections in step 3 are not always applied.
Step 3 in the LMP route to the right of slide is to divide cubical displacements
by radii of curvature. The results are differences in latitude and longitude,
which are then summed. If projected coordinates are required, latitude and
longitude can be converted into Northings and Eastings as is normally done,
that is, with forward projection formulas. Deferring this conversion until
needed is an advantage in the management of directional survey data.
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Radii in the MeridianRadii in the Meridian
W E
S
N
ρρρρ
ρρρρ
Meridian Ellipse
So, what are these radii of the curved Earth?
This slide exhibits the varying radius of curvature in the meridian arc, which is
the N/S ellipse of constant longitude. The radius of curvature in the meridian is
called “rho”. Its length varies as a function of latitude.
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Radii in Prime Vertical and ParallelRadii in Prime Vertical and Parallel
The E/W direction is somewhat more complicated.
The prime vertical is the great circle in the E/W direction. It crosses the
Equator. The radius of curvature in the prime vertical is called “nu” (and it’s
shown here).
The parallel is the small circle of constant latitude in the E/W direction. The
radius of curvature in the parallel of latitude is related to the radius of
curvature in the prime vertical by the cosine of latitude.
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Intermediate QuantitiesIntermediate Quantities
• Given the semi-major axis (a) and
reciprocal of flattening (rf) of the ellipsoid
of the datum of the wellhead location
• Then the semi-minor axis (b) is:
• And the eccentricity squared (e2) is:
)/11/( rfab −=
2222 /)( abae −=
This slide and the next exhibit the formulas for the radii of Earth curvature,
which we’ll touch on only briefly since they are documented in the paper.
Every well has a surface location in latitude and longitude. Latitude and
longitude are referenced to a geodetic datum. Every datum is associated with
an ellipsoid. Every ellipsoid has a semi-major axis (“a”) and a reciprocal of
flattening (“rf”). Given these quantities we can compute two other
intermediate quantities, the semi-minor axis (“b”) and the eccentricity squared
of the ellipsoid (“e-squared”).
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Radii of Earth CurvatureRadii of Earth Curvature
• Radius of curvature in the meridian (ρ) is:
• The prime vertical radius (ν) is:
• Radius of curvature in the parallel is:
23
222 )sin1/()1( φρ eea −−⋅=
21
22 )sin1/( φν ea −=
φν cos⋅
Given the latitude (“phi”) at every station along the wellbore, these simple
formulas give the three radii of curvature discussed previously, “rho”, “nu”
and the radius of curvature in the parallel.
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LMP in PseudocodeLMP in Pseudocode
1. Divide N/S departure by the radius of
curvature in the meridian minus TVD
2. Divide E/W departure by the radius of
curvature in the prime vertical minus
TVD and again by the cosine of
latitude (φ)
3. Results are changes in latitude and
longitude in radians respectively
So, how are these concepts implemented? LMP is expressed in pseudocode as
follows:
First, divide the N/S displacement by the computed radius of curvature in the
meridian minus TVD. Subtracting TVD from the Earth radius accomplishes
depth amplification. TVD is assumed to be referenced to the geoid, if not to the
ellipsoid.
Second, divide the E/W displacement by the computed radius of curvature in
the prime vertical minus TVD and again by the cosine of geodetic latitude.
Finally, these results are changes in latitude and longitude respectively
expressed in radians.
And we're done!
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LMP in MatlabLMP in Matlabrad=180/pi; % Degrees in a radian
A=6378249.145; % A is semi-major axis
RF=293.465; % RF is reciprocal of flattening
B=A*(1-1/RF); % B is semi-minor axis
E2=1-B^2/A^2; % E2 is eccentricity squared
load data.dat; % Load the data
len=length(data); % Count number of survey stations
Edeplast=data(1,6); Ndeplast=data(1,5); % Grab the departures
latlast=data(1,7); lonlast=data(1,8); % Grab the surface geographicals
for dex=1:len % Cycle through all the stations
LATRAD=latlast/rad; % Degrees to radians
R=A*(1-E2)/(sqrt(1-E2*sin(LATRAD)^2)^3); % Radius in the meridian
N=A/sqrt(1-E2*sin(LATRAD)^2); % Radius in the prime vertical
TVD=data(dex,4); % Grab the TVD
Edep=data(dex,6); Ndep=data(dex,5); % Grab the departures
dEdep=Edep-Edeplast; dNdep=Ndep-Ndeplast; % Difference the departures
Edeplast=Edep; Ndeplast=Ndep; % Save the current departures
dlat=rad*dNdep/(R-TVD); % Delta geographicals with LMP ...
dlon=rad*dEdep/(N-TVD)/cos(LATRAD); % ... this is the main idea!
lat=latlast+dlat; lon=lonlast+dlon; % Increment geographicals
latlast=lat; lonlast=lon; % Save geographicals for next loop
data(dex,9)=lat; data(dex,10)=lon; % Store the geographicals in data
end
This slide, which exhibits LMP in Matlab code, is also documented in the
paper, so we’ll not dwell long upon it, either.
The code is on the left. Comments are on the right.
It is worth noting, however, that all the heavy lifting is done by only four lines
of code, those highlighted in yellow. Two radii of curvature are computed here.
The displacements are divided by the radii minus TVD here.
The rest is bookkeeping.
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SummarySummary
• If grid-north and other mapping corrections
are not applied, positioning errors result
– In general, these corrections are not applied
• Grid north is an unobservable mathematical
abstraction, a cause of data-management
mistakes, and best avoided
• LMP is accurate and computationally
efficient, and it avoids the use of grid north
In summary, LMP is a simple, computationally efficient algorithm that
eliminates 5 of the 7 mistakes cited by Williamson and Wilson. The two
remaining mistakes pertain to the datum of the surface and target locations.
They must be prevented by prudent geodetic practice whether LMP is used or
not.
An additional benefit of LMP is that neither grid north nor convergence of the
meridians is needed for the computation of geodetic coordinates. This
eliminates a common source of mistakes in the management of directional-well
survey data.
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